To build a PSQL client container built upon the GCP Cloud SQL Alpine image make a Dockerfile similar to this:
FROM google/cloud-sdk:alpine
RUN apk add --no-cache postgresql-client
RUN addgroup postgres && adduser postgres -G postgres -D -H
RUN wget https://dl.google.com/cloudsql/cloud_sql_proxy.linux.amd64 -O cloud_sql_proxy
RUN chmod +x cloud_sql_proxy
USER postgres
ENTRYPOINT ["psql"]